Wikibooksβ

Trigonometry/Selected Angles Reference

Note: Some values in the table are given in forms that include a radical in the denominator — this is done both to simplify recognition of reciprocal pairs and because the form given in the table is simpler in some sense. Note also that all absolute values of trigonometric functions for remarkable points that are listed in this table are contained in the first quadrant (from 0 to 90° or π2{\displaystyle {\frac {\pi }{2}}} radians, inclusive); all others are deduced by simple symmetries with the horizontal or vertical axis, or by swapping axes (on the trigonometric circle) so that one trigonometric function is also swapped with its co-function.

θ{\displaystyle \theta }(positive)

sin⁡(θ){\displaystyle \sin(\theta )}

cos⁡(θ){\displaystyle \cos(\theta )}

tan⁡(θ){\displaystyle \tan(\theta )}

cot⁡(θ){\displaystyle \cot(\theta )}

sec⁡(θ){\displaystyle \sec(\theta )}

csc⁡(θ){\displaystyle \csc(\theta )}

θ{\displaystyle \theta }(negative)

(degrees)

(radians)

(degrees)

(radians)

0∘{\displaystyle 0^{\circ }}

0{\displaystyle 0}

0{\displaystyle 0}

1{\displaystyle 1}

0{\displaystyle 0}

not
defined

1{\displaystyle 1}

not
defined

−360∘{\displaystyle -360^{\circ }}

−2π{\displaystyle -2\pi }

15∘{\displaystyle 15^{\circ }}

π12{\displaystyle {\frac {\pi }{12}}}

6−24{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

6+24{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

2−3{\displaystyle 2-{\sqrt {3}}}

2+3{\displaystyle 2+{\sqrt {3}}}

6−2{\displaystyle {\sqrt {6}}-{\sqrt {2}}}

6+2{\displaystyle {\sqrt {6}}+{\sqrt {2}}}

−345∘{\displaystyle -345^{\circ }}

−13π12{\displaystyle -{\frac {13\pi }{12}}}

22.5∘{\displaystyle 22.5^{\circ }}

π8{\displaystyle {\frac {\pi }{8}}}

2−22{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

2+22{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

2−1{\displaystyle {\sqrt {2}}-1}

2+1{\displaystyle {\sqrt {2}}+1}

4−22{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}

4+22{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}

−337.5∘{\displaystyle -337.5^{\circ }}

−15π8{\displaystyle -{\frac {15\pi }{8}}}

30∘{\displaystyle 30^{\circ }}

π6{\displaystyle {\frac {\pi }{6}}}

12{\displaystyle {\frac {1}{2}}}

32{\displaystyle {\frac {\sqrt {3}}{2}}}

13{\displaystyle {\frac {1}{\sqrt {3}}}}

3{\displaystyle {\sqrt {3}}}

23{\displaystyle {\frac {2}{\sqrt {3}}}}

2{\displaystyle 2}

−330∘{\displaystyle -330^{\circ }}

−11π6{\displaystyle -{\frac {11\pi }{6}}}

45∘{\displaystyle 45^{\circ }}

π4{\displaystyle {\frac {\pi }{4}}}

12{\displaystyle {\frac {1}{\sqrt {2}}}}

1{\displaystyle 1}

2{\displaystyle {\sqrt {2}}}

−315∘{\displaystyle -315^{\circ }}

−7π4{\displaystyle -{\frac {7\pi }{4}}}

60∘{\displaystyle 60^{\circ }}

π3{\displaystyle {\frac {\pi }{3}}}

32{\displaystyle {\frac {\sqrt {3}}{2}}}

12{\displaystyle {\frac {1}{2}}}

3{\displaystyle {\sqrt {3}}}

13{\displaystyle {\frac {1}{\sqrt {3}}}}

2{\displaystyle 2}

23{\displaystyle {\frac {2}{\sqrt {3}}}}

−300∘{\displaystyle -300^{\circ }}

−5π3{\displaystyle -{\frac {5\pi }{3}}}

67.5∘{\displaystyle 67.5^{\circ }}

3π8{\displaystyle {\frac {3\pi }{8}}}

2+22{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

2−22{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

2+1{\displaystyle {\sqrt {2}}+1}

2−1{\displaystyle {\sqrt {2}}-1}

4+22{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}

4−22{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}

−292.5∘{\displaystyle -292.5^{\circ }}

−11π8{\displaystyle -{\frac {11\pi }{8}}}

75∘{\displaystyle 75^{\circ }}

5π12{\displaystyle {\frac {5\pi }{12}}}

6+24{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

6−24{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

2+3{\displaystyle 2+{\sqrt {3}}}

2−3{\displaystyle 2-{\sqrt {3}}}

6+2{\displaystyle {\sqrt {6}}+{\sqrt {2}}}

6−2{\displaystyle {\sqrt {6}}-{\sqrt {2}}}

−285∘{\displaystyle -285^{\circ }}

−19π12{\displaystyle -{\frac {19\pi }{12}}}

90∘{\displaystyle 90^{\circ }}

π2{\displaystyle {\frac {\pi }{2}}}

1{\displaystyle 1}

0{\displaystyle 0}

not
defined

0{\displaystyle 0}

not
defined

1{\displaystyle 1}

−270∘{\displaystyle -270^{\circ }}

−3π2{\displaystyle -{\frac {3\pi }{2}}}

105∘{\displaystyle 105^{\circ }}

7π12{\displaystyle {\frac {7\pi }{12}}}

6+24{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−6−24{\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

−2−3{\displaystyle -2-{\sqrt {3}}}

−2+3{\displaystyle -2+{\sqrt {3}}}

−6−2{\displaystyle -{\sqrt {6}}-{\sqrt {2}}}

6−2{\displaystyle {\sqrt {6}}-{\sqrt {2}}}

−255∘{\displaystyle -255^{\circ }}

−17π12{\displaystyle -{\frac {17\pi }{12}}}

112.5∘{\displaystyle 112.5^{\circ }}

5π8{\displaystyle {\frac {5\pi }{8}}}

2+22{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

−2−22{\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

−2−1{\displaystyle -{\sqrt {2}}-1}

−2+1{\displaystyle -{\sqrt {2}}+1}

−4+22{\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}

4−22{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}

−247.5∘{\displaystyle -247.5^{\circ }}

−11π8{\displaystyle -{\frac {11\pi }{8}}}

120∘{\displaystyle 120^{\circ }}

2π3{\displaystyle {\frac {2\pi }{3}}}

32{\displaystyle {\frac {\sqrt {3}}{2}}}

−12{\displaystyle -{\frac {1}{2}}}

−3{\displaystyle -{\sqrt {3}}}

−13{\displaystyle -{\frac {1}{\sqrt {3}}}}

−2{\displaystyle -2}

23{\displaystyle {\frac {2}{\sqrt {3}}}}

−240∘{\displaystyle -240^{\circ }}

−4π3{\displaystyle -{\frac {4\pi }{3}}}

135∘{\displaystyle 135^{\circ }}

3π4{\displaystyle {\frac {3\pi }{4}}}

12{\displaystyle {\frac {1}{\sqrt {2}}}}

−12{\displaystyle -{\frac {1}{\sqrt {2}}}}

−1{\displaystyle -1}

−2{\displaystyle -{\sqrt {2}}}

2{\displaystyle {\sqrt {2}}}

−225∘{\displaystyle -225^{\circ }}

−5π4{\displaystyle -{\frac {5\pi }{4}}}

150∘{\displaystyle 150^{\circ }}

5π6{\displaystyle {\frac {5\pi }{6}}}

12{\displaystyle {\frac {1}{2}}}

−32{\displaystyle -{\frac {\sqrt {3}}{2}}}

−13{\displaystyle -{\frac {1}{\sqrt {3}}}}

−3{\displaystyle -{\sqrt {3}}}

−23{\displaystyle -{\frac {2}{\sqrt {3}}}}

2{\displaystyle 2}

−210∘{\displaystyle -210^{\circ }}

−7π6{\displaystyle -{\frac {7\pi }{6}}}

157.5∘{\displaystyle 157.5^{\circ }}

7π8{\displaystyle {\frac {7\pi }{8}}}

2−22{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

−2+22{\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

−2+1{\displaystyle -{\sqrt {2}}+1}

−2−1{\displaystyle -{\sqrt {2}}-1}

−4−22{\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}

4+22{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}

−202.5∘{\displaystyle -202.5^{\circ }}

−9π8{\displaystyle -{\frac {9\pi }{8}}}

165∘{\displaystyle 165^{\circ }}

11π12{\displaystyle {\frac {11\pi }{12}}}

6−24{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

−6+24{\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−2+3{\displaystyle -2+{\sqrt {3}}}

−2−3{\displaystyle -2-{\sqrt {3}}}

−6+2{\displaystyle -{\sqrt {6}}+{\sqrt {2}}}

6+2{\displaystyle {\sqrt {6}}+{\sqrt {2}}}

−195∘{\displaystyle -195^{\circ }}

−13π12{\displaystyle -{\frac {13\pi }{12}}}

180∘{\displaystyle 180^{\circ }}

π{\displaystyle \pi }

0{\displaystyle 0}

−1{\displaystyle -1}

0{\displaystyle 0}

not
defined

−1{\displaystyle -1}

not
defined

−180∘{\displaystyle -180^{\circ }}

−π{\displaystyle -\pi }

195∘{\displaystyle 195^{\circ }}

13π12{\displaystyle {\frac {13\pi }{12}}}

−6−24{\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

−6+24{\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

2−3{\displaystyle 2-{\sqrt {3}}}

2+3{\displaystyle 2+{\sqrt {3}}}

−6+2{\displaystyle -{\sqrt {6}}+{\sqrt {2}}}

−6−2{\displaystyle -{\sqrt {6}}-{\sqrt {2}}}

−165∘{\displaystyle -165^{\circ }}

−11π12{\displaystyle -{\frac {11\pi }{12}}}

202.5∘{\displaystyle 202.5^{\circ }}

9π8{\displaystyle {\frac {9\pi }{8}}}

−2−22{\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

−2+22{\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

2−1{\displaystyle {\sqrt {2}}-1}

2+1{\displaystyle {\sqrt {2}}+1}

−4−22{\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}

−4+22{\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}

−157.5∘{\displaystyle -157.5^{\circ }}

−7π8{\displaystyle -{\frac {7\pi }{8}}}

210∘{\displaystyle 210^{\circ }}

7π6{\displaystyle {\frac {7\pi }{6}}}

−12{\displaystyle -{\frac {1}{2}}}

−32{\displaystyle -{\frac {\sqrt {3}}{2}}}

13{\displaystyle {\frac {1}{\sqrt {3}}}}

3{\displaystyle {\sqrt {3}}}

−23{\displaystyle -{\frac {2}{\sqrt {3}}}}

−2{\displaystyle -2}

−150∘{\displaystyle -150^{\circ }}

−5π6{\displaystyle -{\frac {5\pi }{6}}}

225∘{\displaystyle 225^{\circ }}

5π4{\displaystyle {\frac {5\pi }{4}}}

−12{\displaystyle -{\frac {1}{\sqrt {2}}}}

1{\displaystyle 1}

−2{\displaystyle -{\sqrt {2}}}

−135∘{\displaystyle -135^{\circ }}

−3π4{\displaystyle -{\frac {3\pi }{4}}}

240∘{\displaystyle 240^{\circ }}

4π3{\displaystyle {\frac {4\pi }{3}}}

−32{\displaystyle -{\frac {\sqrt {3}}{2}}}

−12{\displaystyle -{\frac {1}{2}}}

3{\displaystyle {\sqrt {3}}}

13{\displaystyle {\frac {1}{\sqrt {3}}}}

−2{\displaystyle -2}

−23{\displaystyle -{\frac {2}{\sqrt {3}}}}

−120∘{\displaystyle -120^{\circ }}

−2π3{\displaystyle -{\frac {2\pi }{3}}}

247.5∘{\displaystyle 247.5^{\circ }}

11π8{\displaystyle {\frac {11\pi }{8}}}

−2+22{\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

−2−22{\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

2+1{\displaystyle {\sqrt {2}}+1}

2−1{\displaystyle {\sqrt {2}}-1}

−4+22{\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}

−4−22{\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}

−112.5∘{\displaystyle -112.5^{\circ }}

−5π8{\displaystyle -{\frac {5\pi }{8}}}

255∘{\displaystyle 255^{\circ }}

17π12{\displaystyle {\frac {17\pi }{12}}}

−6+24{\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−6−24{\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

2+3{\displaystyle 2+{\sqrt {3}}}

2−3{\displaystyle 2-{\sqrt {3}}}

−6−2{\displaystyle -{\sqrt {6}}-{\sqrt {2}}}

−6+2{\displaystyle -{\sqrt {6}}+{\sqrt {2}}}

−105∘{\displaystyle -105^{\circ }}

−7π12{\displaystyle -{\frac {7\pi }{12}}}

270∘{\displaystyle 270^{\circ }}

3π2{\displaystyle {\frac {3\pi }{2}}}

−1{\displaystyle -1}

0{\displaystyle 0}

not
defined

0{\displaystyle 0}

not
defined

−1{\displaystyle -1}

−90∘{\displaystyle -90^{\circ }}

−π2{\displaystyle -{\frac {\pi }{2}}}

285∘{\displaystyle 285^{\circ }}

19π12{\displaystyle {\frac {19\pi }{12}}}

−6+24{\displaystyle -{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

6−24{\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

−2−3{\displaystyle -2-{\sqrt {3}}}

−2+3{\displaystyle -2+{\sqrt {3}}}

6+2{\displaystyle {\sqrt {6}}+{\sqrt {2}}}

−6+2{\displaystyle -{\sqrt {6}}+{\sqrt {2}}}

−75∘{\displaystyle -75^{\circ }}

−5π12{\displaystyle -{\frac {5\pi }{12}}}

292.5∘{\displaystyle 292.5^{\circ }}

11π8{\displaystyle {\frac {11\pi }{8}}}

−2+22{\displaystyle -{\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

2−22{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

−2−1{\displaystyle -{\sqrt {2}}-1}

−2+1{\displaystyle -{\sqrt {2}}+1}

4+22{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}

−4−22{\displaystyle -{\sqrt {4-2{\sqrt {2}}}}}

−67.5∘{\displaystyle -67.5^{\circ }}

−3π8{\displaystyle -{\frac {3\pi }{8}}}

300∘{\displaystyle 300^{\circ }}

5π3{\displaystyle {\frac {5\pi }{3}}}

−32{\displaystyle -{\frac {\sqrt {3}}{2}}}

12{\displaystyle {\frac {1}{2}}}

−3{\displaystyle -{\sqrt {3}}}

−13{\displaystyle -{\frac {1}{\sqrt {3}}}}

2{\displaystyle 2}

−23{\displaystyle -{\frac {2}{\sqrt {3}}}}

−60∘{\displaystyle -60^{\circ }}

−π3{\displaystyle -{\frac {\pi }{3}}}

315∘{\displaystyle 315^{\circ }}

7π4{\displaystyle {\frac {7\pi }{4}}}

−12{\displaystyle -{\frac {1}{\sqrt {2}}}}

12{\displaystyle {\frac {1}{\sqrt {2}}}}

−1{\displaystyle -1}

2{\displaystyle {\sqrt {2}}}

−2{\displaystyle -{\sqrt {2}}}

−45∘{\displaystyle -45^{\circ }}

−π4{\displaystyle -{\frac {\pi }{4}}}

330∘{\displaystyle 330^{\circ }}

11π6{\displaystyle {\frac {11\pi }{6}}}

−12{\displaystyle -{\frac {1}{2}}}

32{\displaystyle {\frac {\sqrt {3}}{2}}}

−13{\displaystyle -{\frac {1}{\sqrt {3}}}}

−3{\displaystyle -{\sqrt {3}}}

23{\displaystyle {\frac {2}{\sqrt {3}}}}

−2{\displaystyle -2}

−30∘{\displaystyle -30^{\circ }}

−π6{\displaystyle -{\frac {\pi }{6}}}

337.5∘{\displaystyle 337.5^{\circ }}

15π8{\displaystyle {\frac {15\pi }{8}}}

−2−22{\displaystyle -{\frac {\sqrt {2-{\sqrt {2}}}}{2}}}

2+22{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}

−2+1{\displaystyle -{\sqrt {2}}+1}

−2−1{\displaystyle -{\sqrt {2}}-1}

4−22{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}

−4+22{\displaystyle -{\sqrt {4+2{\sqrt {2}}}}}

−22.5∘{\displaystyle -22.5^{\circ }}

−π8{\displaystyle -{\frac {\pi }{8}}}

345∘{\displaystyle 345^{\circ }}

13π12{\displaystyle {\frac {13\pi }{12}}}

−6−24{\displaystyle -{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}

6+24{\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}

−2+3{\displaystyle -2+{\sqrt {3}}}

−2−3{\displaystyle -2-{\sqrt {3}}}

6−2{\displaystyle {\sqrt {6}}-{\sqrt {2}}}

−6−2{\displaystyle -{\sqrt {6}}-{\sqrt {2}}}

−15∘{\displaystyle -15^{\circ }}

−π12{\displaystyle -{\frac {\pi }{12}}}

360∘{\displaystyle 360^{\circ }}

2π{\displaystyle 2\pi }

0{\displaystyle 0}

1{\displaystyle 1}

0{\displaystyle 0}

not
defined

1{\displaystyle 1}

not
defined

0∘{\displaystyle 0^{\circ }}

0{\displaystyle 0}

Notice that for certain values of x{\displaystyle x} , the tangent, cotangent, secant, and cosecant functions are undefined. This is because these functions are defined as sin⁡(x)cos⁡(x){\displaystyle {\frac {\sin(x)}{\cos(x)}}} , cos⁡(x)sin⁡(x){\displaystyle {\frac {\cos(x)}{\sin(x)}}} , 1cos⁡(x){\displaystyle {\frac {1}{\cos(x)}}} , and 1sin⁡(x){\displaystyle {\frac {1}{\sin(x)}}} , respectively. Since an expression is undefined if it contains division by 0, the functions are therefore undefined at angle measures where the denominator (the sine or cosine of x{\displaystyle x} , depending on the trigonometric function) is equal to 0. For example, the tangent function for 90º (π/2{\displaystyle \pi /2\,} radians) is equivalent to sin⁡(π2)cos⁡(π2){\displaystyle {\frac {\sin \left({\frac {\pi }{2}}\right)}{\cos \left({\frac {\pi }{2}}\right)}}} , or 10{\displaystyle {\frac {1}{0}}} , which is an undefined value.